Does the Old Testament apply to Christians today or is it only the New Testament that really matters?
Of course the Old Testament applies to Christians today. It comprises over three fourths of our Bibles. Our job is to understand how it applies, not to decide if it applies. Evidence that it applies today is that for the early church the Old Testament was the only Bible they had and they certainly used it. Jesus quoted extensively from the Old Testament, as did the New Testament writers. Take a look at the book of Hebrews and look in the footnotes at the number of quotes and uses of the Old Testament. Also, if you read the early church fathers they quoted extensively from the Old Testament even as they were just beginning to quote the New Testament. The letter of Clement of Rome is more or less a series of quotes from the Old Testament, with comments between the quotes on how it applies to Christians.
Two passages which point to the use of the Old Testament for Christians are found in Romans 15:4 (for endurance and encouragement) and 1 Corinthians 10:11 (as examples and warnings for us). In 2 Tim 3:16 Paul tells us that all Scripture is inspired by God and useful for admonishing, teaching and rebuking. He is referring principally to the Old Testament here because when he wrote the New Testament was not even a well-defined collection of writings.
Having said all that, when we read the Old Testament, we read it differently in some ways from the New Testament. The examples from history are obviously instructive. The advice in Proverbs applies to us equally as they did to the Jews. The inspiration for our relationship with God in Psalms is more or less what the Jews before Jesus would have gotten from these passages. However, there are differences. For example, the laws received at Mt. Sinai do not apply to us in the same way they applied to the Jews. Jesus fulfilled the requirements of the Law and we are no longer required to observe these laws for salvation. This is the tone of the entire book of Galatians. Like Paul said, “Now that faith has come, we are no longer under the supervision of the law (presumably of Moses) (Galatians 3:25). We do not need to observe the Sabbath, to go to a priest when we get a skin disease, to offer various burnt, guilt and sin offerings, or to avoid eating certain kinds of food (Mark 7:19). Nevertheless, we can learn, even from these commandments. Each of them has a spiritual meaning with analogous teaching for us today as Christians, even if we do not observe the literal commands. For example, in Leviticus 17:11 it says that “the life is in the blood.” Even though we are not forbidden to eat what is forbidden in this passage, we certainly can see the spiritual application to the statement that our life is in the blood.
As a general rule, we can study the Old Testament to know God–to learn good theology, but we do not read the Old Testament as a source of doctrines and commandments for Christians. It is not as simple as this and we can find many exceptions, but that is a rule we can use to begin to understand how to study the Old Testament. It is a source of encouragement, warning, teaching, advice, theology, evidence of inspiration and more, but many of its commandments, especially in Numbers, Leviticus and Dueteronomy are not requirements for us as followers of Jesus.
I hope this helps.